arXiv:1712.05454v1 [math.SP] 14 Dec 2017 ieetao,taevco,cmlxHdmr MSC: Hadamard 2010 complex vector, trace differentiator, Keywords: Implications generaliz a matrices. discussed. and circulant Malamud on are an Tyaglov to research and vectors, due Kushel trace result to a due and of result matrix differentiators proof Hadamard on alternate complex result an a provide a by prove similarity case we via ditionally important comp realizable of spectra via variety and realizable ces, a spectra Sule˘ımanova spectra, for and spectra, established conjecture, Ciarlet Johnson’s is work, this conjecture, th In that Monov’s conjectured realizable. further be must Johnson points nonnegative. are list izable ceietesetao nrws ongtv arcs oesp More (herein matrices. multi-set nonnegative finite entrywise a of spectra the acterize Introduction 1. mat nonnegative entrywise an of n spectrum complex the of is the multiset that it finite conjectured if Monov A realizable matrices. called nonnegative is character entrywise to is of (NIEP) tra problem eigenvalue inverse nonnegative The Abstract [email protected] [email protected] rpitsbitdt ierAgbaadisApiain S Applications its and Algebra Linear to submitted Preprint http://amber.thrall.me/ nteraiaiiyo h rtclpit farealizable a of points critical the of realizability the On b ✩ aa .Hoover L. Sarah ∗ lrd nttt fTcnlg,10W nvriyBv. M Blvd., University W. 150 Technology, of Institute Florida c h longstanding The mi addresses: Email author. Corresponding hswr a upre yNFAward NSF by supported was work This URL: iiino niern n ahmtc,Uiest fWas of University Mathematics, and Engineering of Division http://faculty.washington.edu/pietrop/ a aitnClee 9 olg ilR. lno,N,13323, NY, Clinton, Rd., Hill College 198 College, Hamilton ongtv nes ievlepolm Sule˘ımanova spectrum, problem, eigenvalue inverse Nonnegative 52,1A8 54,30C15 15B48, 15A18, 15A29, [email protected] a,1 apsWyN,Btel A901 USA 98011, WA Bothell, NE, Way Campus ailA McCormick A. Daniel , ongtv nes ievleproblem eigenvalue inverse nonnegative AbrR Thrall) R. (Amber Dne .McCormick), A. (Daniel list AbrR Thrall) R. (Amber = Λ ) k th mmnso h ito rtclpit fareal- a of points critical of list the of -moments .Thrall R. { λ list 1 λ , . . . , SrhL Hoover), L. (Sarah DMS-1460699 ✩ c,1 b,1 n } [email protected] Per Paparella), (Pietro itoPaparella Pietro , fcmlxnmes h NIEP the numbers, complex of . lore L 20,USA 32901, FL, elbourne, igo ohl,18115 Bothell, hington Per Paparella), (Pietro NE)i ochar- to is (NIEP) pebr1,2018 18, eptember ito critical of list e cfial,given ecifically, USA c,1, consequently no matri- anion z h spec- the ize ∗ including s s tto it use d Amber , o further for to fa of ation umbers Ad- . rix. asks for necessary and sufficient conditions such that Λ is the spectrum of an n-by-n entrywise-. If Λ is the spectrum of a nonnegative matrix A, then Λ is called realizable and A is a called realizing matrix (for Λ). It is well-known that if Λ = λ1,...,λn is realizable, then Λ must satisfy the following conditions: { }

Λ := λ¯ ,..., λ¯n = Λ; (1) { 1 } ρ(Λ) := max λi Λ; (2) 1 i n ≤ ≤ {| |} ∈ n k sk(Λ) := λ 0, k N; and (3) i ≥ ∀ ∈ i=1 m Xm 1 s (Λ) n − skm(Λ), k,m N. (4) k ≤ ∀ ∈ It is worth noting that these conditions are not independent: Loewy and London [24] showed that the moment condition (3) implies the self-conjugacy condition (1); Friedland [12, Theorem 1] showed that the eventual nonnegativity of the moments implies the spectral radius condition (2); and if the trace is nonnega- tive, i.e., if s1(Λ) 0, then the J-LL condition (4) (established independently by Johnson [15] and≥ by Loewy and London [24]) implies the moment condition since 1 k sk(Λ) k 1 s1 (Λ) 0. ≥ n − ≥ Thus, the J-LL condition and the nonnegativity of the trace imply the self- conjugacy, spectral radius, and moment conditions. There is another necessary condition due to Holtz [13], which was shown to be independent of J-LL, but for brevity, it will not be considered here (for more information regarding the NIEP, there are several articles [2, 6, 11, 17, 26, 32] and monographs [26, 1] that discuss the problem). n Let Λ = λ ,...,λn C (n 2) be a list, let p(z) = (z λi), and { 1 } ⊂ ≥ i=1 − let Λ′ = µ1,...,µn 1 denote the zeros of p′. Monov [27] posed the following conjecture.{ − } Q

Conjecture 1.1 (Monov). If Λ is realizable, then sk(Λ′) 0 for all k N. ≥ ∈ The following is due to Johnson [16].

Conjecture 1.2 (Johnson). If Λ is realizable, then Λ′ is realizable. It is clear that Johnson’s conjecture implies Monov’s conjecture. Cronin and Laffey [8] established Johnson’s conjecture in the cases when n 4, or when ≤ n 6 and s1(Λ) = 0, . ≤Note that the converse of Conjecture 1.2 is not true; consider the list Λ = 1, 1, 2/3, 2/3, 2/3 and the polynomial p(z) = (z 1)2(z +2/3)3. Then { − 4 − 2 − } − p′(z)=5z 5z 20/27z+20/27,Λ′ = 1, 1/3, 2/3, 2/3 ,andΛ′ is realizable [24]. However,− it− is well-known that Λ is{ not realizable− − (see,} e.g., Friedland [12] and Paparella and Taylor [28] for a more general result). Note that there are instances when the constant of integration can be selected so that the zeros of an antiderivative form a realizable set (see Section 6).

2 In this work, we investigate and constructively prove Conjecture 1.2 for a variety of cases. First, we examine the kth moments of the critical points of a realizable list and establish a sufficient condition for realizability via companion matrices. Next, we explore the d- as a construction for a realizing matrix of the critical points and establish a sufficient condition that includes Ciarlet spectra [7]. We then study differentiators following the work of Pereira [29] and show that Johnson’s conjecture holds for spectra realizable via complex Hadamard similarities. Along the way, we provide an alternate proof to a result of Malamud [25] and extend the results on circulant matrices by Kushel and Tyaglov [20]. Additionally, we use these results to provide a new proof of a classical theorem on the interlacing roots and critical points for polynomials. Finally, implications for further research are discussed.

2. Notation

n Unless otherwise stated, Λ := λ ,...,λn C (n 2), p(t) := (t { 1 } ⊂ ≥ i=1 − λi) C[t], and Λ′ := µ C p′(µ)=0 = µ1,...,µn 1 . Furthermore, we ∈ { ∈ | } { − } Q refer to the critical points of p, i.e., the zeros of p′, as the critical points of Λ. For n N, we let n denote the set 1,...,n . ∈ h i { } We let Mm n(F) denote the set of m-by-n matrices with entries from a field × F. In the case when m = n, Mm n(F) is abbreviated to Mn(F). × For A Mn(F), we let σ(A) denote the spectrum (i.e., list of eigenvalues) of A; we let ∈A denote the ith principal submatrix of A, i.e., A is the (n 1)- (i) (i) − by-(n 1) matrix obtained by deleting the ith-row and ith-column of A; and we let τ(A−) denote the normalized trace of A, i.e., τ(A) := (1/n) tr A. th We let diag (λ1,...,λn) denote the whose i diagonal entry is λi. For A Mm(F) and B Mn(F), the direct sum of A and B, denoted A B, is the matrix∈ ∈ ⊕ A 0 Mm n(F). 0 B ∈ ×   The of A Mm n(F) and B Mp q(F), denoted A B, is the (mp)-by-(nq) ∈ × ∈ × ⊗

a11B ... a1nB . . .  . .. .  . a B ... a B  m1 mn    3. Preliminary Results

Proposition 3.1. If Λ is realizable, then Λ′ = Λ′. Proof. Follows from the fact that the derivative of a real polynomial is real.

Proposition 3.2. If Λ is realizable, then s (Λ′) 0. 1 ≥

3 Proof. Since 1 1 s (Λ) = s (Λ′), n 1 n 1 1 − it follows that n 1 s (Λ′)= − s (Λ) 0. 1 n 1 ≥ Monov [27, Proposition 2.3] provides a formula that relates the moments of a list with the moments of its critical points. Theorem 3.3 (Monov’s moment formula). If

s1(Λ) n 0 ... 0 2s (Λ) s (Λ) n . . . 0 2 1 . . . . dk(Λ) := . . . . ,

(k 1)sk 1(Λ) sk 2(Λ) sk 3(Λ) ... n − − − − ksk(Λ) sk 1(Λ) sk 2(Λ) ... s1(Λ) − −

then 1 k sk(Λ′)= sk(Λ) + dk(Λ). (5) −n   Corollary 3.4. If Λ is realizable, then s (Λ′) 0. 2 ≥ Proof. A straightforward application of Theorem 3.3 yields

n 2 1 2 s (Λ′)= − s (Λ) + s (Λ) 0. 2 n 2 n2 1 ≥ With this we see a particular case of the J-LL condition satisfied. Corollary 3.5. If Λ is a list, then

s2(Λ) ns (Λ) (6) 1 ≤ 2 if and only if 2 s (Λ′) (n 1)s (Λ′). 1 ≤ − 2 Proof. Following (5), n 2 n 2 s2(Λ) ns (Λ) 0 − s (Λ) − s2(Λ) 1 ≤ 2 ⇐⇒ ≤ n 2 − n2 1 n 2 1 1 n 2 0 − s (Λ) + s2(Λ) s2(Λ) − s2(Λ) ⇐⇒ ≤ n 2 n2 1 − n2 1 − n2 1 n 2 1 n 1 0 − s (Λ) + s2(Λ) − s2(Λ) ⇐⇒ ≤ n 2 n2 1 − n2 1 n 2 1 n 1 2 0 (n 1) − s (Λ) + s2(Λ) − s (Λ) ⇐⇒ ≤ − n 2 n2 1 − n 1     2 0 (n 1)s (Λ′) s (Λ′) ⇐⇒ ≤ − 2 − 1 2 s (Λ′) (n 1)s (Λ′). ⇐⇒ 1 ≤ − 2

4 Inequality (6) is important for the realizability of lists with three elements [24, Theorem 2], as well as generalized Sule˘ımanova spectra [21], which we define in the sequel. A real list Λ is called a Sule˘ımanova spectrum if s1(Λ) 0 and Λ contains exactly one nonnegative entry. A list Λ is called a generalized≥ Sule˘ımanova spectrum when s1(Λ) 0 and Λ contains exactly one entry with nonnegative real part. ≥ Friedland [12] and Perfect [30] proved that all Sule˘ımanova spectra are re- alizable by a companion matrix. In particular, they showed that if Λ is a Sule˘ımanova spectrum and

n n n n k p(t)= (t λi)= t + an kt − , − − i=1 k Y X=1 then ak 0 for every k n . Consequently, the companion matrix ≤ ∈ h i

0 a0 C(p) := − , a := a1 an 1 ⊤ In 1 a ··· −  − −    is nonnegative. Note that the class of spectra realizable via companion matrices contains spectra other than (generalized) Sule˘ımanova spectra (e.g., the nth roots of unity for n 4). ≥ Theorem 3.6. If Λ is a list such that C(p) is nonnegative, then Λ′ is realizable. Furthermore, Λ′ is realizable by a companion matrix. Proof. By hypothesis, n n n k p(t)= t + an kt − , − k X=1 1 with ak 0 for every k n . Thus, the non-monic coefficients of p′ are ≤ ∈ h i n nonpositive, i.e., C(p′/n) 0. ≥ Remark 3.7. If p is a polynomial such that C(p) is nonnegative, then, for 1 k n 1, the zeros of p(k) are realizable. Moreover, if the constants of integration≤ ≤ are− chosen to be nonpositve, then the roots of its successive anti-derivatives are also realizable (see Section 6). Theorem 3.8. If Λ is a realizable generalized Sule˘ımanova spectrum such that s1(Λ) = 0, then Λ′ is realizable. Proof. Laffey and Smigocˇ [21] showed that Λ is realizable by a matrix of the form C + αI, where C is a trace-zero companion matrix and α is a nonnegative scalar. Since 0 = s1(Λ) = tr(C + αI)=tr C + α tr I = nα, it follows that α = 0. Thus, Λ is realizable by a trace-zero companion matrix, so Λ′ is realizable. The following theorem is a classical result on the relationship between poly- nomial and its critical points when its zeros are real.

5 Theorem 3.9. Let p be a polynomial with real roots λ1,...,λn and critical points µ1,...,µn 1, each listed in descending order. Then the critical points of p interlace the roots− of p, i.e.,

λ1 µ1 λ2 ...λn 1 µn 1 λn. ≥ ≥ ≥ − ≥ − ≥ Theorem 3.9 can be proved via Rolle’s theorem and some simple root count- ing arguments, but the details are somewhat cumbersome. Instead, we defer this proof to Stection 5 to highlight the utility of the results presented there. While Theorem 3.6 proves that the critical points of Sule˘ımanova spectra are realizable, the following is worth noting.

Theorem 3.10. If Λ is a Sule˘ımanova spectrum, then Λ′ is a Sule˘ımanova spectrum. Proof. Since p has all real roots, by Theorem 3.9 the critical points must all be real and interlace the zeros. Thus, there is at most one nonnegative critical point. Furthermore, by Theorem 3.2, s1(Λ′) 0, so there must be at least one nonnegative critical point. ≥

4. The D-Companion Matrix

Just as the roots of a polynomial can be studied with a companion matrix, Cheung and Ng [4] introduced the d-companion matrix to study the critical points of a polynomial. n Definition 4.1 (d-companion matrix). Let p(t)= (t λi) be a polynomial i=1 − of degree n 2 and let D := diag (λ2,...,λn) Mn 1(C). If I and J denote the (n 1)-by-(≥ n 1) identity and all-ones matrices∈ Q − respectively, then the the matrix− − 1 λ A = D I J + 1 J − n n   is called a d-companion matrix of p. Notice that, after relabeling the zeros of p, there are n possible choices for the d-companion matrix. Cheung and Ng [4, Theorem 1.1] showed that the spectrum of any d-companion matrix of a polynomial coincides with its multiset of critical points. A straightforward calculation reveals that

1 (n 1)λi+1 + λ1 i = j aij = − (7) n λ λi i = j. ( 1 − +1 6

Remark 4.2. If Λ R is realizable, with λ = ρ(Λ), then (7) reveals that Λ′ is ⊂ 1 the spectrum of a , i.e., Λ′ is realizable by a matrix of the form A = B αI, with B 0 and α 0. − ≥ ≥ Ciarlet [7] showed that the following condition is sufficient for realizability.

6 Theorem 4.3 (Ciarlet). Let Λ= λ ,...,λn R such that λ λn. If { 1 }⊂ 1 ≥···≥ nλn + λ 0, then Λ is realizable. 1 ≥ From the d-companion matrix, we derive a similar necessary condition for the realizability of critical points.

Theorem 4.4. Let Λ = λ ,...,λn R be realizable and suppose that λ { 1 } ⊂ 1 ≥ λn. If (n 1)λn + λ 0, then Λ′ is realizable. ···≥ − 1 ≥ Proof. Immediate in view of (7). Corollary 4.5. The critical points of a Ciarlet spectrum are realizable. Although not directly applicable to the primary problem considered here, given a polynomial with complex roots, a construction of the d-companion ma- trix with real entries may be useful elsewhere.

Theorem 4.6. If Λ= λ ,...,λm,µ , µ¯ ,...,µn, µ¯n , where Im(λi)=0, i { 1 1 1 } ∀ ∈ m and Im(µi) = 0, i n , then the spectrum of the matrix (whose entries harei real) 6 ∀ ∈ h i 1 λ B I K + 1 K, − m +2n m +2n   is the critical points of Λ, where

λ2 n . Re µi Im µi B =  ..  , ⊕ Im µi Re µi ! λ i=1 −   m M   J(m 1) (m 1) G K = − × − , G⊤ C  

G = J(m 1) n √2 0 , − × ⊗ and  

2 0 C = Jn n . × ⊗ 0 0   Proof. If 1 1 1 V := , √2 i i  −  then V is unitary and, for any complex number z,

Re z Im z V diag (z, z¯) V = . ∗ Im z Re z −  n If A is the the d-companion matrix U := Im m i V , then U is unitary × ⊕ =1 and UAU ∗ is the desired matrix. L

7 5. Differentiators and Complex Hadamard Matrices

Pereira [29], following the work of Davis [9], studied differentiators and in- troduced the concept of a trace vector to solve several unsolved conjectures in the geometry of polynomials. Although Pereira studied differentiators and trace vectors in the setting of linear operators on finite dimensional Hilbert spaces, we only consider these objects for matrices and summarize some of his results in this setting. n Definition 5.1. For A Mn(C) and a unit vector z C , let P = P (z) := ∈ ∈ I zz∗ Mn(C). If B := P AP P Cn (the matrix B is called the compression of A−onto P∈Cn), then P is called a| differentiator (of A) if 1 p (t)= p′ (t), B n A where pM denotes the the characteristic polynomial of M Mn(C). ∈ n Definition 5.2. Let A Mn(C) and z C . Then z is called a trace vector k k ∈ ∈ (of A) if z∗A z = τ(A ), for every nonnegative integer k. Remark 5.3. Corresponding to the case when k = 0, it is clear that all trace vectors have unit length. The following result is due to Pereira. n Theorem 5.4 (Pereira [29, Theorem 2.5]). If A Mn(C) and z C is a unit vector, then P is a differentiator of A if and only∈ if z is a trace vector∈ of A. In addition, Pereira [29, Theorem 2.10] shows that every matrix possesses a trace vector, i.e., every matrix possesses a differentiator.

Example 5.5. If ei is a trace vector of A Mn(C), where ei (1 i n) denotes th Cn ∈ ≤ ≤ the i canonical basis vector of , and P = I eiei⊤, then the compression of n − A onto P C is A(i).

Proposition 5.6. Let A Mn(C) and let z be a trace vector of A. If α is a complex number such that ∈α =1, then αz is a trace vector of A. | | Proof. For any nonnegative integer k, notice that

k k k k (αz)∗A (αz)=¯ααz∗A z = z∗A z = τ(A ). A stronger result is available for diagonal matrices. n 1 Lemma 5.7. Let D Mn(C) be a diagonal matrix and z C . If zi = ∈ ∈ | | √n for all i n , then z is a trace vector of D. ∈ h i Proof. If k is a nonnegative integer, then n n n k k 2 k 1 k k z∗D z = zizid = zi d = d = τ(D ), ii | | ii n ii i=1 i=1 i=1 X X X i.e., z is a trace vector of D.

8 Remark 5.8. The converse of Lemma 5.7 fails; indeed, if D is a diagonal matrix, then e ∗ e d i D i = ii = τ(D), i n . √n √n n ∀ ∈ h i     Although not relevant to this work, we note the following result. Proposition 5.9. If z Cn is a trace vector of every n-by-n diagonal matrix, 1 ∈ then zi = for all i n . | | √n ∈ h i Proof. Suppose that z Cn is a trace vector of all diagonal matrices and let ∈ i n . If D = eie⊤, then ∈ h i i n 1 k k k 2 = τ(D )= z∗D z = zj zjd = zi , n jj | | j=1 X and the result is established upon taking square-roots throughout.

Lemma 5.10. If Q = αS, with α C 0 and S GLn(C), then for A 1 1 ∈ \{ } ∈ ∈ Mn(C), QAQ− = SAS− . 1 1 α 1 1 Proof. QAQ− = (αS)A(αS)− = α SAS− = SAS− . Malamud [25, Proposition 4.2] provides the following result on the relation- ship between a polynomial and its critical points. Proposition 5.11 (Malamud). If p is a polynomial, then there is a A such that σ(A)=Λ and σ(A(n))=Λ′.

In particular, Malamud shows that if A = UDU ∗, where D = diag (λ1,...,λn) and U is a satisfying 1 unj = , j n , (8) | | √n ∀ ∈ h i then the matrix A satisfies the conclusion of Proposition 5.11. Following the work of Pereira, we provide an alternate proof to the above result and generalize it to any principal submatrix of A.

Theorem 5.12. Let Λ= λ1,...,λn be a list, and let U Mn(C) be a unitary { } 1 ∈ matrix such that for some i n , uij = for all j n . If A = UDU ∗, ∈ h i | | √n ∈ h i where D = diag (λ1,...,λn), then σ A(i) =Λ′.

Proof. Suffices to show that ei is a trace  vector of A. Notice that U ∗ei is the entrywise complex conjugate of the ith-row of U. By hypothesis, each element 1 of this row has modulus √n , thus, by Lemma 5.7, U ∗ei is a trace vector of D. Moreover, if k is any nonnegative integer, then

k k k 1 1 e∗A e = e∗ UD U ∗ e = (U ∗e )∗D (U ∗e )= tr D = tr A = τ(A, i i i i i i n n  i.e., ei is trace vector for A.

9 A proof of Proposition 5.11 is immediate once a unitary matrix is provided that satisfies property (8). Recall that an n-by-n matrix H is called a (of order n) if hij 1 and HH⊤ = nI. A matrix H is called a ∈ {± } (of order n) if hij = 1 and HH∗ = nI. Notice | | 1 that for any complex Hadamard matrix H Mn(C), the matrix U := H is ∈ √n unitary. For a fixed positive integer n and the complex scalar ω := exp( 2πi/n), the matrix − 11 1 ... 1 2 n 1 1 ω ω ... ω −  2 4 2(n 1) 1 ω ω ... ω − F = Fn := , . . . . .  ......   2  1 ωn 1 ω2(n 1) ... ω(n 1)   − − −    is called the discrete (DFT) matrix (of order n). As is well- known, the DFT matrices are complex Hadamard matrices. Since ωk = ω k = 1k = 1 for every integer k, it follows that every row of the unitary| | matrix| | 1 U := √n Fn satisfies the condition required in Proposition 5.11. Furthermore, if Λ = λ ,...,λn is a list and A = UDU ∗, D = diag (λ ,...,λn), then, { 1 } 1 following Theorem 5.12, σ(A )=Λ′, for every i n . (i) ∈ h i For S GLn(C), the set ∈ n 1 (S) := x C SDS− 0,D = diag (x ,...,xn) C { ∈ | ≥ 1 } is called the spectracone of S [18, 19]. It is known that the spectracones corre- sponding to Hadamard matrices comprise a large class of realizable real spectra [19]. Furthermore, spectracones corresponding to DFT matrices comprise a large class of realizable nonreal spectra [18].

Corollary 5.13. Let Λ= λ1,...,λn be a list and H be a complex Hadamard { } 1 matrix. If A = UDU ∗ 0, where U := H and D := diag (λ ,...,λn), then ≥ √n 1 Λ′ is realizable. In particular, A(i) is a realizing matrix for Λ′. Remark 5.14. There is further evidence that Conjectrure 1.2 is true; recall that Boyle and Handelman [3] found necessary and sufficient conditions for Λ=Λ˜ 0,..., 0 to be the spectrum of a nonnegative matrix. Laffey [22] gave a construction∪{ for} a realizing n-by-n matrix X. Laffey et al. [23] showed that ˜ en is a trace vector of X, thus the critical points of Λ are realizable by X(n). A matrix of the form

c c cn 1 2 ··· cn c1 cn 1 C =  . . ··· .−  (9) . . .   c c c   2 3 ··· 1    is called a or circulant. It is well-known that a matrix C is a circulant if and only if there is a diagonal matrix D such that C = UDU ∗, where

10 1 U := √n Fn [10, Theorems 3.2.2 and 3.2.3]. Kushel and Tyaglov [20, Theorem 2.1] showed that if p is the characteristic polynomial of C and Λ = σ(C), then σ(C(n))=Λ′. However, Theorem 5.12 yields a more general result with a simpler proof.

Corollary 5.15. If C is defined as in (9), Λ = σ(C), then σ(C(i))=Λ′, for every i n . ∈ h i Proof. Immediate in view of Theorem 5.12 and Lemma 5.10. We conclude this section with a proof of Theorem 3.9.

1 Proof of Theorem 3.9. Let Λ be a real list. Set A = H diag (λ1,...,λn) H− , where H is any complex Hadamard matrix. By Lemma 5.10, A is a unitary similarity of a real diagonal matrix, so it is Hermitian. Furthermore, by Theorem 5.12, σ A(i) = Λ′, for every i n . By the Cauchy interlacing theorem [14, Theorem 4.3.17], Λ interlaces Λ.∈ h i  ′

6. Concluding Remarks

Since Johnson’s conjecture is stronger than Monov’s conjecture, we have shown both Johnson’s and Monov’s conjecture for Sule˘ımaonva spectra, Cia- rlet spectra, spectra realizable by companion matrices, and spectra realizable by complex Hadamard Similarities. In the general case, Johnson’s conjecture remains open for n 5. Since the J-LL and≥ trace conditions imply the moment condition, proving that J-LL is satisfied for the critical points of realizable lists is a worthwhile avenue for investigation, as this would prove Monov’s conjecture. Cheung and Ng [5], recognize that other constructions of a d-companion matrix exist. They provide methods for constructing such matrices which have already proven fruitful in the study of circulant matrices [20]. Future research on finding alternative constructions to the d-companion matrix may be critical to solving this conjecture. It is simple to show that, for spectra realizable via companion matrices, there exists an anti-derivative with roots that are realizable as well. Notice that if n n n k p(t) = t + k=1 an kt − , where each ak 0, then any anti-derivative is of the form − ≤ P n 1 n+1 1 n k+1 P (t)= t + an kt − + c, ak 0, n +1 n k +1 − ≤ k X=1 − Notice that (n + 1)P is monic and its companion matrix is nonnegative if and only if c is nonpositive. Thus, there are infinitely-many anti-derivatives whose zeros form a realizable list. Bhat and Mukherjee [31] introduced integrators as an analog to differen- tiatos. While every matrix has a differentiator [29], the existence of an integra- tor is neither guaranteed nor unique when an integrator is available. An explicit construction of an integrator is an avenue worthy of further exploration.

11 Acknowledgements

The authors would like to thank the National Science Foundation for funding the REU program and the University of Washington Bothell for hosting the research program. In particular, we would like to thank Professors Jennifer McLoud-Mann and Casey Mann for their efforts.

References

[1] A. Berman and R. J. Plemmons. Nonnegative matrices in the mathematical sciences, volume 9 of Classics in Applied Mathematics. Society for Indus- trial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. Revised reprint of the 1979 original. [2] A. Borobia. On the nonnegative eigenvalue problem. Linear Algebra Appl., 223/224:131–140, 1995. Special issue honoring Miroslav Fiedler and Vlas- timil Pt´ak. [3] M. Boyle and D. Handelman. The spectra of nonnegative matrices via symbolic dynamics. Ann. of Math. (2), 133(2):249–316, 1991. [4] W. S. Cheung and T. W. Ng. A companion matrix approach to the study of zeros and critical points of a polynomial. J. Math. Anal. Appl., 319(2):690– 707, 2006. [5] W. S. Cheung and T. W. Ng. Relationship between the zeros of two poly- nomials. Linear Algebra Appl., 432(1):107–115, 2010. [6] M. T. Chu. Inverse eigenvalue problems. SIAM Rev., 40(1):1–39, 1998. [7] P. G. Ciarlet. Some results in the theory of nonnegative matrices. Linear Algebra and Appl., 1(1):139–152, 1968. [8] A. G. Cronin and T. J. Laffey. Perturbations on constructible lists pre- serving realizability in the NIEP and questions of Monov. Linear Algebra Appl., 445:206–222, 2014. [9] C. Davis. Eigenvalues of compressions. Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.), 3 (51):3–5, 1959. [10] P. J. Davis. Circulant matrices. John Wiley & Sons, New York-Chichester- Brisbane, 1979. A Wiley-Interscience Publication, Pure and Applied Math- ematics. [11] P. D. Egleston, T. D. Lenker, and S. K. Narayan. The nonnegative in- verse eigenvalue problem. Linear Algebra Appl., 379:475–490, 2004. Tenth Conference of the International Linear Algebra Society. [12] S. Friedland. On an inverse problem for nonnegative and eventually non- negative matrices. Israel J. Math., 29(1):43–60, 1978.

12 [13] O. Holtz. M-matrices satisfy newton’s inequalities. Proc. Amer. Math. Soc., 133(3):711–717, 2004. [14] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, New York, NY, USA, 2nd edition, 2012. [15] C. R. Johnson. Row stochastic matrices similar to doubly stochastic ma- trices. Linear and Multilinear Algebra, 10(2):113–130, 1981. [16] C. R. Johnson. Personal communication, 2015. [17] C. R. Johnson, C. Mariju´an, P. Paparella, and M. Pisonero. The NIEP. Operator Theory: Advances and Applications. To appear, https://arxiv.org/abs/1703.10992. [18] C. R. Johnson and P. Paparella. Perron similarities and the nonnegative inverse eigenvalue problem. In preparation. [19] C. R. Johnson and P. Paparella. Perron spectratopes and the real non- negative inverse eigenvalue problem. Linear Algebra Appl., 493:281–300, 2016. [20] O. Kushel and M. Tyaglov. Circulants and critical points of polynomials. J. Math. Anal. Appl., 439(2):634–650, 2016. [21] H. Laffey, Thomas J. Smigoc. Nonnegative realization of spectra having negative real parts. Linear Algebra and Appl., 416:148–159, 2006. [22] T. J. Laffey. A constructive version of the Boyle-Handelman theorem on the spectra of nonnegative matrices. Linear Algebra Appl., 436(6):1701–1709, 2012. [23] T. J. Laffey, R. Loewy, and H. Smigoc.ˇ Power series with positive co- efficients arising from the characteristic polynomials of positive matrices. Math. Ann., 364(1-2):687–707, 2016. [24] R. Loewy and D. London. A note on an inverse problem for nonnegative matrices. Linear and Multilinear Algebra, 6(1):83–90, 1978/79. [25] S. M. Malamud. Inverse spectral problem for normal matrices and the Gauss-Lucas theorem. Trans. Amer. Math. Soc., 357(10):4043–4064, 2005. [26] H. Minc. Nonnegative matrices. Wiley-Interscience Series in Discrete Math- ematics and Optimization. John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. [27] V. V. Monov. A family of symmetric polynomials of the eigenvalues of a matrix. Linear Algebra Appl., 429(8-9):2199–2208, 2008. [28] P. Paparella and G. K. Taylor. The volume of the trace-nonnegative poly- tope via the Irwin-Hall distribution. Minnesota Journal of Undergraduate Mathematics. To appear, https://arxiv.org/abs/1511.03700.

13 [29] R. Pereira. Differentiators and the geometry of polynomials. J. Math. Anal. Appl., 285(1):336–348, 2003. [30] H. Perfect. Methods of constructing certain stochastic matrices. Duke Math. J., 20(3):395–404, 09 1953. [31] B. V. Rajarama Bhat and M. Mukherjee. Integrators of matrices. Linear Algebra Appl., 426(1):71–82, 2007. [32] R. L. Soto. Nonnegative inverse eigenvalue problem. In H. A. Yasser, editor, Linear Algebra - Theorems and Applications, chapter 05. InTech, Rijeka, 2012.

14